A method for incoherent reflection modulation

Abstract

The present application provides a method of non-coherent reflection modulation, which utilizes active reflection elements in a smart reflection surface RIS, and the information bits are carried by the arrangement matrix of the reflection pattern and the constellation symbol. At the same time, the energy signal and differential technology are used to enable NRM to work without any channel state information CSI on the transmitter, RIS or receiver. Simulation results show that when the number of noise and the number of channels are the same, compared with the existing differential reflection modulation DRM, NRM has a 9dB bit error rate BER performance advantage and stronger noise resistance when the number of reflection elements is 8.

Description

Technical Field

[0001] The present application relates to the technical field of coherent modulation, and in particular to a method of incoherent reflection modulation. Background Art

[0002] In recent years, reconfigurable intelligent surfaces (RIS) composed of a large number of low-cost and passive components have attracted widespread attention in the communications community. RIS-based transmission is completely different from existing technologies. It can modify the propagation environment through integrated intelligent programmable components to improve signal quality. The existing literature on RIS-assisted systems is very extensive, especially in the field of fully passive RIS reflection systems. Passive RIS systems adjust the impact signal by using low-power electronic components. These components can perform unique functions such as reflecting, amplifying, and absorbing the incident signal with a specific reflection coefficient, thereby changing the amplitude and phase of the incident signal, offsetting the impact of the channel, and reflecting the incident signal to the desired direction, thereby improving the BER performance of the system with lower power consumption. At present, passive RIS is widely used in most RIS-based communication systems. Although this saves energy consumption, the performance improvement of RIS communication systems is relatively limited.

[0003] In the past decade, a lot of research has been done on index modulation (IM) technology. This is one of the revolutionary transmission paradigms that uses the index of variables such as antennas, relays, antenna patterns, time slots, etc. in typical wireless communication systems to transmit additional information bits. The presence of a large number of elements on RIS makes it very suitable for the idea of ​​IM, which has led to a large number of systems that combine RIS and IM, such as mapping information bits on RIS elements or using RIS to reflect only the index of a specific receiving antenna for signal transmission, further improving the transmission rate of information bits. Spatial modulation, as a special form of index modulation, has also been combined with RIS in many ways.

[0004] The combination of index modulation and RIS greatly improves the transmission efficiency of the system, but the existing RIS system needs to obtain accurate instantaneous / statistical channel state information (CSI) for detection during transmission. In RIS-based communication systems, due to the presence of a large number of RIS elements, channel estimation from the transmitter to RIS and from RIS to the receiver is a very complex task. Although some work has been done to avoid the problem of accurate channel estimation, such as using imperfect channel estimation, channel estimation itself still consumes time and frequency resources. Especially in the next generation 6G mobile network and industrial Internet of Things, the presence of a large number of unmanned high-speed moving nodes causes various complex characteristics such as frequency selective fading in the wireless channel. Therefore, a large number of pilots are required for effective information transmission, which reduces the proportion of effective information. In response to the above problems, the current way to avoid channel estimation is to use the idea of ​​differential spatial modulation technology in the RIS system. Existing research combines RIS with differential spatial modulation and proposes a differential reflective modulation (DRM) system to avoid channel estimation for a large number of RIS elements. Unlike traditional spatial modulation systems, the DRM system's array matrix arranges the RIS reflection pattern and constellation symbols together. In each time slot, the RIS will generate a different reflection pattern to reflect a single transmitted signal, greatly improving the system's performance.

[0005] Although the DRM system has good BER performance, there are still some problems. On the one hand, the dimension of the permutation matrix of the DRM system is highly correlated with the bit error rate, that is, when the dimension of the permutation matrix is ​​enlarged, the number of reflection modes used increases, resulting in a partial decrease in the BER performance of the system; when the number of system channels and noise is consistent, the system BER performance will be greatly reduced. On the other hand, in the uplink channel, if the transmitter is in motion and the channel state changes dramatically, it is difficult to achieve the quasi-static channel conditions required by the differential. Therefore, in response to the above problems, this patent proposes a noncoherent reflection modulation (NRM) system. In the NRM system, active RIS is used for signal reflection and forwarding. At the transmitter where the channel changes dramatically, the information bit is first encoded into the permutation matrix of the reflection mode and the amplitude of the transmission signal. At the relatively stable RIS end, the amplitude information of the signal is converted into phase information for transmission, ensuring that the receiving end can obtain better BER performance even when the number of receiving antennas is small. This design makes use of the large number of RIS elements to make up for the poor BER performance of the amplitude signal when there are fewer receiving antennas, so that the system can perform relatively accurate detection even when the channel from the transmitter to the RIS is in a fast fading condition. At the same time, a large number of RIS elements are used for transmit diversity from the RIS to the receiving end to ensure system performance. Simulation results show that when the number of noise and the number of channels are the same, compared with the existing DRM system, the NRM scheme proposed in this paper has a BER performance advantage of about 3dB when the number of RIS elements is small (N=2), and the performance advantage expands as N increases. At the same time, as the number of time slots increases, the loss of NRM in BER performance is much smaller than that of DRM. Summary of the invention

[0006] In view of the defects in the above background technology, the present application provides a method of incoherent reflection modulation, which involves spatial modulation technology, index modulation technology, differential technology, energy modulation technology, and intelligent reflection surface, and is mainly used to improve the transmission performance of the communication system. The present application adopts the following technical solutions:

[0007] A method for incoherent reflection modulation, the method comprising the following steps:

[0008] Step 1: Divide the input bits b of each transmission block into two parts, b1 and b2, where the input b1 bits are used to select L constellation symbols, and b2 bits are used to select the legal permutation matrix P t ;

[0009] Step 2: Select L constellation symbols based on b1 input bits, and select a legal permutation matrix P based on b2 input bits t ;

[0010] Step 3: Perform M-PAM signal modulation on b1 input bits, and perform RIS reflection mode and symbol mapping;

[0011] Step 4: At the active RIS, the index of the arriving signal is determined and converted into the corresponding PSK signal according to the index to obtain the PSK symbol vector. The symbol vector is diagonalized and the PSK signal is moved to the permutation matrix to obtain the forwarded data matrix X. t ;

[0012] Step 5: Data matrix X t Perform the difference to get the transmitted signal matrix S t ;

[0013] Step 6: The receiver uses the maximum likelihood detector to demodulate and obtain the estimated matrix The decoded information is used to restore the original information bits.

[0014] Furthermore, in step 1, the calculation formula of the input bit b is as follows:

[0015]

[0016] Where L is the number of time slots, b1 = Llog2M,

[0017] Furthermore, in step 2, the matrix P is arranged t There is only one non-zero element in each row and column of P t ∈P L×L , where P is the set of all possible permutation matrices; from L! P t Select one of As the legal permutation matrix P t , arranging the constellation symbols and reflection patterns.

[0018] Furthermore, the reflection pattern is a diagonal matrix Φ i ∈C N×N , where i = 1, ..., L; Φ i The kth element on the diagonal is represented by (Φ i ) kk =a k exp(jθ k ), where k = 1, ..., N, a k ∈{0, 1} represents the switch state of the kth RIS element, θ k represents the reflection phase of the kth RIS element on the diagonal, in NRM θ k The value of is 0.

[0019] Further, in step 3, M-PAM signal modulation is performed, including: during transmission, b1 bits are first mapped into L M-PAM symbols x l , l∈{1,2,...,L}, b1 bits reach RIS through channel h1, and the signal at RIS is The calculation formula is as follows:

[0020]

[0021] where z1∈C N×1 is a complex Gaussian noise vector with mean 0 and variance N0.

[0022] Furthermore, in step 3, RIS reflection mode and symbol mapping are performed, including:

[0023] Let X PAM =diag([x1,x2,...,x L ]), diag(.) is the diagonalization function, and h1 is expanded to H1=[h1,h1,...,h1]∈C N×L , we get the following matrix-vector expression

[0024] Y R =H1X PAM +Z1 (4)

[0025] where H1∈C N×L is a product with a mean of 0 and a variance of σ 2 = 1 Rayleigh fading channel matrix, Z1∈C N×L is a complex Gaussian noise matrix with mean 0 and variance N0.

[0026] Furthermore, in step 4, the data matrix X t The calculation formula is as follows:

[0027]

[0028] in represents the Kronecker product, I LN ∈C LN×LN For the unit array.

[0029] Furthermore, in step 4, the index of the arriving signal is determined and converted into a corresponding PSK signal according to the index, including:

[0030] Remove the receiving signal The superscript l of y R As a simplified expression, assuming that the transmitted signal is x, in order to detect the M-PAM signal index, it is necessary to maximize the conditional probability density function f(y R |x)

[0031]

[0032] The maximum likelihood method can be expressed as

[0033]

[0034] Let α=|x| 2 +σ 2 The above formula is transformed into

[0035]

[0036] in

[0037] Furthermore, in step 5, the signal matrix S is transmitted t The calculation formula is as follows:

[0038] S t =S t-1 X t (11)

[0039] where S0∈C LN×LN Usually a unit array.

[0040] Further, in step 6, the maximum likelihood detector demodulation includes:

[0041] Through the following matrix

[0042]

[0043]

[0044] Assuming Channel For a quasi-static channel, the received signals at time t and time t-1 are

[0045]

[0046]

[0047] Where Z t YesN r ×N complex Gaussian matrix, whose element z nt Indicates that the Gaussian white noise with a mean of 0 and a variance of N0 is received by the nth receiving antenna in the tth time slot. The received signal y j represents the signal on each receiving antenna, j∈{1,2,...,N r};

[0048] The received signal at time t can be rewritten as

[0049] Yt =Y t-1 X t +Z t -Z t-1 X t (16)

[0050] The maximum likelihood detector demodulation calculation formula is as follows

[0051]

[0052] Where X NRM represents the set of all possible data matrices, represents the F-norm.

[0053] Through the embodiments of the present application, the following technical effects can be obtained: the present application utilizes RIS to make up for the defect of poor performance of energy signals when there are fewer antennas, and utilizes differential technology to avoid detecting channel state information; the transmit matrix uses different rules from DSM, but retains the inherent advantages of the DSM system. Compared with the existing DRM system, NRM has a BER performance advantage of about 6dB and 9dB when the number of reflection elements is 4 and 8, respectively. Through comprehensive computer simulation, the superiority of the proposed NRM scheme over the existing system is proved. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] In order to more clearly illustrate the technical solutions in the embodiments of the present application, a brief introduction will be given below to the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0055] Figure 1 is a schematic diagram of a non-coherent modulation system;

[0056] Figure 2 It is a schematic diagram of the modulation / demodulation process of the non-coherent modulation system;

[0057] Figure 3 For the incoherent modulation system and the differential reflection modulation system, L = 2, N r =3, schematic diagram of BER comparison for BPSK modulation and different numbers of RIS elements N;

[0058] Figure 4 For the NRM system, L = 2, N r =3, schematic diagram of BER comparison under different N and different modulation modes;

[0059] Figure 5 For the NRM system, K = 2, N r=3, L=2, BPSK modulation and different N when the BER performance of the ML algorithm and the fast incoherent algorithm are compared at the RIS. DETAILED DESCRIPTION

[0060] In order to make the purpose, technical solution and advantages of the embodiments of the present application clearer, the technical solution in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0061] The present application proposes a method of non-coherent reflection modulation, which utilizes active reflection elements in a smart reflection surface (RIS), wherein information bits are carried by an arrangement matrix of reflection patterns and constellation symbols, and energy signals and differential techniques are utilized, so that NRM can operate without any channel state information (CSI) on a transmitter, RIS or receiver.

[0062] Figure 1 The non-coherent modulation system NRM (Noncoherent Reflecting Modulation, NRM) includes a transmitter, a smart reflective surface RIS (Reconfigurable Intelligent Surface, RIS), and a receiver, wherein the number of transmitting antennas is N. t , the number of receiving antennas is N r , RIS has N reflection elements; h1∈C N×1 , They represent the channel vector from the transmitting antenna to the RIS and the channel matrix from the RIS to the receiving end, respectively. Each element of the channel matrix is ​​a zero-mean and variance σ 2 =1 complex Gaussian random variable;

[0063] In the NRM system, a constellation symbol with M energy levels or phases is used, that is, an M-PSK signal. The energy level of the M-PAM signal is Where p1 = 0, the average signal energy Where E ave is the average signal power; the mth optimal constellation energy level is expressed as

[0064]

[0065] Where N0 is the variance of complex Gaussian noise;

[0066] Optimal constellation energy level It can be obtained by solving the following formula

[0067]

[0068] Where SNR = E ave / N0 represents the signal-to-noise ratio, and v is the summation sign index. Figure 2 The modulation / demodulation process diagram of the non-coherent modulation system includes the following steps:

[0069] Step 1: Divide the input bits b of each transmission block into two parts, b1 and b2, where the input b1 bits are used to select L constellation symbols, and b2 bits are used to select the permutation matrix P t ;

[0070] In step 1, the input bit b is calculated as follows:

[0071]

[0072] Where L is the number of time slots, b1 = Llog2M,

[0073] Step 2: Select L constellation symbols based on b1 input bits, and select a legal permutation matrix P based on b2 input bits t ;

[0074] In step 2, the permutation matrix P t There is only one non-zero element in each row and column of P t ∈P L×L , where P is the set of all possible permutation matrices; from L! P t Select one of As the legal permutation matrix P t , arranging constellation symbols and reflection patterns;

[0075] The reflection pattern is a diagonal matrix Φ i ∈C N×N , where i = 1, ..., L; Φ i The kth element on the diagonal is represented by (Φ i ) kk =a k exp(jθ k ), where k = 1, ..., N, a k ∈{0, 1} represents the switch state of the kth RIS element, θ k represents the reflection phase of the kth RIS element on the diagonal, in NRM θ k The value of is 0.

[0076] Step 3: Perform M-PAM signal modulation on b1 input bits, and perform RIS reflection mode and symbol mapping;

[0077] In step 3, M-PAM signal modulation is performed, including:

[0078] During transmission, b1 bits are first mapped into L M-PAM symbols x l , l∈{1,2,...,L}, b1 bits reach RIS through channel h1, and the signal at RIS is The calculation formula is as follows:

[0079]

[0080] where z1∈C N×1 is a complex Gaussian noise vector with mean 0 and variance N0.

[0081] In step 3, RIS reflection mode and symbol mapping are performed, including:

[0082] Let X PAM =diag([x1,x2,...,x L ]), diag(.) is the diagonalization function, and h1 is expanded to H1=[h1,h1,...,h1]∈C N×L , we get the following matrix-vector expression

[0083] Y R =H1X PAM +Z1 (4)

[0084] where H1∈C N×L is a product with a mean of 0 and a variance of σ 2 = 1 Rayleigh fading channel matrix, Z1∈C N×L is a complex Gaussian noise matrix with mean 0 and variance N0.

[0085] Step 4: Determine the index of the arriving signal at the active RIS and convert it into the corresponding PSK signal according to the index to obtain the PSK symbol vector T t =[x′1,x′2,...,x′ L ] T , diagonalize the symbol vector, and then move the PSK signal to the permutation matrix to obtain the forwarded data matrix X t ;

[0086] In step 4, the data matrix X t The calculation formula is as follows:

[0087]

[0088] in represents the Kronecker product, I LN ∈C LN×LN For the unit array.

[0089] In step 4, the index of the arriving signal is determined and converted into a corresponding PSK signal according to the index, including:

[0090] Remove the receiving signal The superscript l of y R As a simplified expression, assuming that the transmitted signal is x, in order to detect the M-PAM signal index, it is necessary to maximize the conditional probability density function f(y R |x)

[0091]

[0092] The maximum likelihood method can be expressed as

[0093]

[0094] Let α=|x| 2 +σ 2 The above formula is transformed into

[0095]

[0096] in

[0097] In order to further simplify the RIS decoding and forwarding process and reduce its energy consumption, the following fast non-coherent algorithm is used for signal decoding and forwarding at RIS, which is achieved by quantizing the decision area. Consider And the decision threshold set

[0098] The optimal estimate of the transmitted signal using a fast incoherent detector is determined as follows:

[0099]

[0100] Step 5: Data matrix X t Perform the difference to get the transmitted signal matrix S t ;

[0101] In step 5, the signal matrix S is transmitted t The calculation formula is as follows:

[0102] S t =S t-1 X t (11)

[0103] where S0∈C LN×LN Usually a unit array.

[0104] Step 6: The receiver uses the maximum likelihood detector to demodulate and obtain the estimated matrix The decoded information is used to restore the original information bits.

[0105] Through the following matrix

[0106]

[0107]

[0108] Assuming Channel For a quasi-static channel, the received signals at time t and time t-1 are

[0109]

[0110]

[0111] Where Z t YesN r ×N complex Gaussian matrix, whose element z nt Indicates that the Gaussian white noise with a mean of 0 and a variance of N0 is received by the nth receiving antenna in the tth time slot. The received signal y j represents the signal on each receiving antenna, j∈{1,2,...,N r};

[0112] The received signal at time t can be rewritten as

[0113] Y t =Y t-1 X t +Z t -Z t-1 X t (16)

[0114] The maximum likelihood detector demodulation calculation formula is as follows

[0115]

[0116] Where X NRM represents the set of all possible data matrices, represents the F-norm.

[0117] Figure 3 The bit error rate performance of DRM and NRM is compared when the number of RIS elements N is 2, 4 and 8, where L = 2, N r=3, the modulation mode is BPSK modulation (M=2). It can be seen that when N changes, the DRM system changes little, and adding reflection elements cannot significantly improve the BER performance of DRM. Since the NRM system uses PAM signals, it is very sensitive to the number of reflection elements. The reception accuracy of the PAM signal is positively correlated with the number of N. When N=2, the performance gap between DRM and NRM is small, only less than 3dB. When N=8, the performance of NRM has been greatly improved. Its BER is more than 6dB higher than that of NRM when N=2, and about 8dB higher than that of DRM when N=8. Obviously, as N increases, the gap will increase further, which shows the performance advantage of the NRM system.

[0118] like Figure 4 The figure shows the BER performance comparison of the NRM system under different RIS element numbers and modulation modes, where L = 2, N r =3. From the figure, we can see that the BER performance of the NRM system drops sharply with the increase of the modulation order. The BER performance gap between BPSK (M=2) and QPSK (M=4) is about 9dB. On the other hand, since the NRM system has no limit on the number of RIS elements, the BER performance of the system also increases sharply with the increase of N. When N=50 and BPSK modulation is used, the BER of the NRM system at SNR=3 reaches 10 -5 ; At low SNR, when N=50 and QPSK, its BER performance is much lower than that when N=4 and BPSK, but when the SNR increases to 9, the performance of both is less than 1dB. It can be seen that the increase in the number of RIS elements N can not only greatly improve the BER performance of the system, but also significantly offset the performance loss caused by the increase in the modulation order.

[0119] Figure 5 For K = 2, N r =3, L = 2, BPSK and different N RIS using ML algorithm and fast incoherent algorithm BER performance was compared. From the figure, we can see that when N is small or the signal-to-noise ratio is low, the BER performance of the two methods is basically close. But when N is relatively large and the signal-to-noise ratio is high, the BER performance of the two methods begins to differ. For example, when N = 8 and SNR = 9dB, the performance of the fast incoherent algorithm is 0.5dB lower than that of the ML algorithm. However, since the fast incoherent algorithm only needs to perform comparison operations after determining the threshold, the computational complexity is much lower than that of the ML algorithm that needs to traverse, so the fast incoherent algorithm is still better than the ML algorithm overall.

[0120] In summary, the present application utilizes RIS to make up for the defect of poor performance of energy signal when there are fewer antennas, and utilizes differential technology to avoid detecting channel state information; the transmit matrix uses different rules from DSM, but retains the inherent advantages of the DSM system. Compared with the existing DRM system, NRM has a BER performance advantage of about 6dB and 9dB when the number of reflection elements is 4 and 8, respectively. Through comprehensive computer simulation, the superiority of the proposed NRM scheme over the existing system is proved.

[0121] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solution of the present application rather than to limit it. Although the present application has been described in detail through the above preferred embodiments, those skilled in the art should understand that various changes can be made in form and details without departing from the scope defined by the claims of the present application.

Claims

1. A method for incoherent reflection modulation, characterized in that: The method comprises the following steps: Step 1: Divide the input bits b of each transmission block into two parts, b1 and b2, where the input b1 bits are used to select L constellation symbols, and b2 bits are used to select the legal permutation matrix P t ; Step 2: Select L constellation symbols based on b1 input bits, and select a legal permutation matrix P based on b2 input bits t ; Step 3: Perform M-PAM signal modulation on b1 input bits, and perform RIS reflection mode and symbol mapping; Step 4: At the active RIS, the index of the arriving signal is determined and converted into the corresponding PSK signal according to the index to obtain the PSK symbol vector. The symbol vector is diagonalized and the PSK signal is moved to the permutation matrix to obtain the forwarded data matrix X. t ; Step 5: Data matrix X t Perform the difference to get the transmitted signal matrix S t ; Step 6: The receiver uses the maximum likelihood detector to demodulate and obtain the estimated matrix The decoded information is used to restore the original information bits.

2. The method according to claim 1, characterized in that In step 1, the input bit b is calculated as follows: Where L is the number of time slots, M is the number of energy levels, b1 = Llog2M, 3. The method according to claim 1, characterized in that In step 2, the permutation matrix P t There is only one non-zero element in each row and column of P t ∈P L×L , where P is the set of all possible permutation matrices; from L! P t Select one of As the legal permutation matrix P t , arranging the constellation symbols and reflection patterns.

4. The method according to claim 2, characterized in that: The reflection pattern is a diagonal matrix Φ i ∈C N×N , where i = 1, ..., L; Φ i The kth element on the diagonal is represented by (Φ i ) kk =a k exp(jθ k ), where k = 1, ..., N, a k ∈{0, 1} represents the switch state of the kth RIS element, θ k represents the reflection phase of the kth RIS element on the diagonal, in NRM θ k The value of is 0.

5. The method according to claim 1, characterized in that In step 3, M-PAM signal modulation is performed, including: during transmission, b1 bits are first mapped into L M-PAM symbols x l , l∈{1,2,...,L}, b1 bits reach RIS through channel h1, and the signal at RIS is The calculation formula is as follows: where Z1∈C N×L is a complex Gaussian noise vector with mean 0 and variance N0, where N is the number of reflected elements.

6. The method according to claim 1, characterized in that In step 3, RIS reflection mode and symbol mapping are performed, including: Let X PAM =diag([x1,x2,...,x L ]), diag(.) is the diagonalization function, and h1 is expanded to H1=[h1,h1,...,h1]∈C N×L , we get the following matrix-vector expression AND R =H1X PAM +Z1 (4) where H1∈C N×L is a 0-mean, σ 2 = 1 Rayleigh fading channel matrix, Z1∈C N×L is a complex Gaussian noise matrix with mean 0 and variance N0, x L represents L M-PAM symbols mapped to , and h1 represents the channel vector from the transmitting antenna to RIS.

7. The method according to claim 1, characterized in that In step 4, the data matrix X t The calculation formula is as follows: in represents the Kronecker product, I LN ∈C LN×LN is the unit matrix, T t Represents a PSK symbol vector.

8. The method according to claim 1, characterized in that In step 4, the index of the arriving signal is determined and converted into a corresponding PSK signal according to the index, including: Remove the receiving signal The superscript l of y R As a simplified expression, assuming that the transmitted signal is x, in order to detect the M-PAM signal index, it is necessary to maximize the conditional probability density function f(y R |x) The maximum likelihood method can be expressed as Let α=|x| 2 +σ 2 The above formula is transformed into in σ 2 represents variance, p m represents the mth optimal constellation energy level, Indicates the energy level of the M-PAM signal.

9. The method according to claim 1, characterized in that: In step 5, the signal matrix S is transmitted t The calculation formula is as follows: S t =S t-1 X t (11) where S0∈C LN×LN It is usually a unit matrix, and LN represents the order of the matrix.

10. The method according to claim 1, characterized in that In step 6, the maximum likelihood detector demodulates, including: Through the following matrix Among them, N r represents the number of receiving antennas, Φ represents the diagonal matrix, and LN represents the order of the matrix; Assuming Channel For a quasi-static channel, the received signals at time t and time t-1 are Where Z t YesN r ×N complex Gaussian matrix, whose element z nt Indicates that the Gaussian white noise with a mean of 0 and a variance of N0 is received by the nth receiving antenna in the tth time slot. The received signal y j represents the signal on each receiving antenna, j∈{1,2,...,N r }; The received signal at time t can be rewritten as Y t =Y t-1 X t +Z t -Z t-1 X t (16) The calculation formula for maximum likelihood detector demodulation is as follows Where X NRM represents the set of all possible data matrices, represents the F-norm.

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